교과서 질문
In Exercises 35–68, use integration, the Direct Comparison Test, or the Limit Comparison Test to test the integrals for convergence. If more than one method applies, use whatever method you prefer.
∫ from e to e^e of (ln(ln x) dx)
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Ch. 8 - Techniques of Integration
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
8장, 문제 8.7.36
[Technology Exercise] When solving Exercises 33-40, you may need to use a calculator or a computer.
Find, to two decimal places, the areas of the surfaces generated by revolving the curves in Exercises 35 and 36 about the x-axis.
y = x²/4, 0 ≤ x ≤ 2
검증된 단계별 안내1
Identify the formula for the surface area of a solid of revolution about the x-axis:
\[ S = \int_a^b 2\pi y \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx \]
Given the curve \[ y = \frac{x^2}{4} \], find its derivative with respect to \[ x \]:
\[ \frac{dy}{dx} = \frac{d}{dx} \left( \frac{x^2}{4} \right) = \frac{x}{2} \]
Substitute \[ y \] and \[ \frac{dy}{dx} \] into the surface area formula:
\[ S = \int_0^2 2\pi \left( \frac{x^2}{4} \right) \sqrt{1 + \left( \frac{x}{2} \right)^2} \, dx \]
Simplify the expression inside the square root and the integrand:
\[ S = \int_0^2 2\pi \frac{x^2}{4} \sqrt{1 + \frac{x^2}{4}} \, dx = \int_0^2 \frac{\pi x^2}{2} \sqrt{1 + \frac{x^2}{4}} \, dx \]
Evaluate the integral using an appropriate method (such as substitution or numerical integration) to find the surface area, then round the result to two decimal places.
비슷한 문제에 대한 검증된 영상 답변:
이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
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주요 개념
질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.
Surface Area of Revolution
The surface area of a solid formed by revolving a curve around an axis is found using an integral formula. For revolution about the x-axis, the formula involves integrating 2π times the radius (the y-value) times the arc length element. This concept connects geometry with calculus to measure curved surfaces.
추천 영상:
09:07
Example 1: Minimizing Surface Area
Arc Length Element (ds)
The arc length element ds represents a small segment of the curve and is given by √(1 + (dy/dx)²) dx. It accounts for the curve's slope, ensuring the surface area calculation accurately follows the curve's shape rather than just the x-interval.
추천 영상:
06:29Arc Length of Parametric Curves
Definite Integration with Limits
Definite integration calculates the exact accumulated value over an interval, here from x = 0 to x = 2. Applying limits ensures the surface area corresponds precisely to the specified portion of the curve, providing a numerical result to two decimal places.
추천 영상:
05:43Definition of the Definite Integral
관련 실천
교과서 질문
For Exercises 49–52, complete the square before using an appropriate trigonometric substitution.
∫ √(x² + 2x + 2) / (x² + 2x + 1) dx
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교과서 질문
[Technology Exercise] When solving Exercises 33-40, you may need to use a calculator or a computer.
Find, to two decimal places, the areas of the surfaces generated by revolving the curves in Exercises 35 and 36 about the x-axis.
y = sin x, 0 ≤ x ≤ π
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교과서 질문
Use any method to evaluate the integrals in Exercises 15–38. Most will require trigonometric substitutions, but some can be evaluated by other methods.
∫ √(9 - w²) dw / w²
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교과서 질문
Evaluate the integrals in Exercises 23–32.
∫₀^(π/6) √(1 + sin(x)) dx
(Hint: Multiply by √((1 - sin(x)) / (1 - sin(x))))
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교과서 질문
Use any method to evaluate the integrals in Exercises 15–38. Most will require trigonometric substitutions, but some can be evaluated by other methods.
∫ (x dx) / (25 + 4x²)
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